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AP Pre Calc > 3b > Flashcards

3b Flashcards

1
Q

inverse of a function

A

the result of switching the input and output values of a function

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2
Q

the output of an inverse trigonometric function will be…

A

an angle measure

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3
Q

two ways to represent inverse trigonometric functions

A

1) sin^-1
2) arcsin

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4
Q

why are domains of inverse trigonometric functions restricted?

A

they wouldn’t be functions without the restrictions! (wouldn’t pass the vertical line test! (normal trigonometric functions don’t pass horizontal line test))

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5
Q

to have inverses, a function must pass…

A

the horizontal line test

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6
Q

restricted domain of an sin function

A

[-pi/2, pi/2]

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7
Q

restricted domain of an cos function

A

[0, pi]

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8
Q

restricted domain of an tan function

A

(-pi/2, pi/2)

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9
Q

the restricted domain of a function is the ____ of its inverse function

A

range of possible solutions

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10
Q

restricted domains as shaded on unit circle

A

sin ~ Q1, Q4 (right half)
cos ~ Q1, Q2 (top half)
tan~ Q1, Q4 (right half)

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11
Q

after find solutions of an inverse function…

A

double check they are in the allotted domain!

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12
Q

is sin(x)=1/2 equivalent to arcsin(1/2)=x

A

No, both have pi/6 as a solution but due to the restricted domain of the inverse function that is its only solution whereas the sin function has multiple (infinite) solutions.

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13
Q

since trigonometric functions are periodic…

A

they can have infinitely many solutions (if the domain is not restricted)

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14
Q

general solution to sin(x)=1/2

A

x=pi/6+2pik
and
x=5pi/6+2pik
where k is any integer

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15
Q

general solution to cos(x)=1/2

A

x=pi/3+2pik
and
x=5/3+2pik
where k is any integer

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16
Q

general solution to tan(x)=1

A

x=pi/4+pik
where k is any integer

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17
Q

general solution to all sin and cos functions

A

-find solutions for x on first rotation of unit circle
-write an equation for each with +2pik attached
-clarify k is any integer

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18
Q

general solution to all tan functions

A

-find smallest solution for x on first rotation of unit circle
-write an equation with +pik attached to the solution
-clarify k is any integer

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19
Q

steps to solve trig equations

A
  1. isolate the trig function
  2. find the corresponding angles that satisfy the equation
  3. consider the domain restrictions
    –if note add pik or 2pik
  4. write solutions
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20
Q

REMEMBER WITH SQUARE ROOTS

A

solutions can be +/-

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21
Q

(sinx)^n=

A

sin^nx

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22
Q

[] vs ()

A

[] if </> or equal to (includes end values)
() if strictly </> (does not include max and min values)

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23
Q

steps for solving trig inequalities

A
  1. set f(x)=0 and solve
  2. make a sign chart with you solutions (remember domain restrictions!)
  3. test a value in each interval
  4. interpret the sign chart and answer the inequality
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24
Q

</> signs flip when…

A

you divide by a negative number

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25
Q

secant

A

sec=1/cos

26
Q

cosecant

A

csc=1/sin

27
Q

cotangent

A

cot=1/tan=cos/sin

28
Q

Graphs of sec, csc, cot are undefined where…

A

sec, csc, tan hit x axis

29
Q

range of sec

A

(-inf, -1] [1, inf)

30
Q

graph of sec and csc

A

mini parabolas on top of maxs and mins

31
Q

undefined ares of sec graph

A

x=pi/2+pik

32
Q

range of csc

A

(-inf, -1] [1, inf)

33
Q

range of cot

A

(-inf, inf)

33
Q

undefined areas of csc graph

A

x=pik

34
Q

undefined areas of cot graph

A

x=pik

35
Q

cot graph

A

negative version of tan graph shifted over pi/2, so it fits the new asymptotes of pik

36
Q

to find asymptotes of an equation

A

plug x and the transformations that directly affect it into the “x” of the equation and solve so x is by itself

37
Q

to solve csc, cot, sec

A

isolate and change into sin, cos, tan to use unit citcle

38
Q

pythagorean identity

A

cos^2x+sin^2x=1

39
Q

you can change the pythagorean identity

A

into a lot!
–even tan, if you divide by cos!
–to solve for sin^2x or cos^2x

40
Q

sum and diff identities

A

sin(a+/-b)=sinacosb+/-cosasinb
- sign is the same
cos(a+/-b)=cosacosb-/+sinasinb
- sign changes

41
Q

double angle identities

A

sin(2x)=2sinxcosx
cos(2x)=cos^2x-sin^2x=1-2sin^2x=2cos^x-1

42
Q

polar coordinates

A

r, theta
output, input

43
Q

y =

A

sin(theta) * r

44
Q

x =

A

cos(theta) *r

45
Q

r =

A

+/- sqrt (x^2 + y^2)

46
Q

tan(theta) =

A

y/x

47
Q

i =

A

sqrt(-1)

48
Q

complex number rectangular coordinates

A

a+bi = (a,b)

49
Q

complex number polar coordinates

A

(rcos(theta))+i(rsin(theta))

50
Q

r(theta) = asin(theta), +/- a?

A

+ ~ symmetrical to y axis on top
- ~ symmetrical to y axis on bottom

51
Q

r(theta) = acos(theta), +/- a?

A

+ ~ symmetrical to x axis on right
- ~ symmetrical to x axis on left

52
Q

if b is even…

A

graph will have 2 b petals

53
Q

if b is odd…

A

graph will have b petals

54
Q

if r is positive and increasing, the distance between r and the origin is…

A

increasing

55
Q

if r is negative and decreasing, the distance between r and the origin is…

A

increasing

56
Q

if r is positive and decreasing, the distance between r and the origin is…

A

decreasing

57
Q

if r is negative and increasing, the distance between r and the origin is…

A

decreasing

58
Q

relative extrema occur

A

when graph changes from increasing to decreasing or vice versa

59
Q

AROC of a polar function

A

y2-y1/x2-x1
indicates the rate at which the radius is changing per radian

60
Q

estimating with AROC

A

f(x) = known y + AROC (x-known x)

AP Pre Calc (10 decks)

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